现代分析及其应用教程

本书通过度量空间中序列的收敛性讨论了完备性和紧性等问题，并给出了解决相关问题的方法，还阐述了现代分析中的另一种拓扑方法。本书可应用到微分方程和积分方程、线性代数方程组、近似理论、数值分析和量子力学等领域，适合数学本科生、数学教师和其他需要学习一些数学分析知识用于其他领域的读者参考使用。

Preface
1 Prelude to Modern Analysis
1.1 Introduction
1.2 Sets and numbers
1.3 Functions or mappings
1.4 Countability
1.5 Point sets
1.6 Open and closed sets
1.7 Sequences
1.8 Series
1.9 Functions of a real variable
1.10 Uniform convergence
1.11 Some linear algebra
1.12 Setting off
2 Metric Spaces
2.1 Definition of a metric space
2.2 Examples of metric spaces
2.3 Solved problems
2.4 Exercises
2.5 Convergence in a metric space
2.6 Examples on completeness
2.7 Subspace of a metric space
2.8 Solved problems
2.9 Exercises
3 The Fixed Point Theorem and its Applications
3.1 Mappings between metric spaces
3.2 The fixed point theorem
3.3 Applications
3.4 Perturbation mappings
3.5 Exercises
4 Compactness
4.1 Compact sets
4.2 Ascoli's theorem
4.3 Application to approximation theory
4.4 Solved problems
4.5 Exercises
5 Topological Spaces
5.1 Definitions and examples
5.2 Closed sets
5.3 Compact sets
5.4 Continuity in topological spaces
5.5 Homeomorphisms; connectedness
5.6 Solved problems
5.7 Exercises
6 Normed Vector Spaces
6.1 Definition of a normed vector space; examples
6.2 Convergence in normed spaces
6.3 Solved problems
6.4 Exercises
6.5 Finite-dimensional normed vector spaces
6.6 Some approximation theory
6.7 Chebyshev theory
6.8 The Weierstrass approximation theorem
6.9 Solved problems
6.10 Exercises
7 Mappings on Normed Spaces
7.1 Bounded linear mappings
7.2 Norm of an operator
7.3 Functionals
7.4 Solved problems
7.5 Exercises
7.6 Inverse mappings
7.7 Application to integral equations
7.8 Application to numerical analysis
7.9 Exercises
7.10 Unbounded mappings
8 Inner Product Spaces
8.1 Definitions; simple consequences
8.2 Orthonormal vectors
8.3 Least squares approximation
8.4 The Riesz representation theorem
8.5 Solved problems
8.6 Exercises
9 Hilbert Space
9.1 Definition of Hilbert space
9.2 The adjoint operator
9.3 Separability
9.4 Solved problems
9.5 Exercises
9.6 Complete orthonormal sets; generalised Fourier series
9.7 Hilbert space isomorphism
9.8 Exercises
Bibliography
Selected Solutions
Index
编辑手记